Approximations of the population Fisher information matrix- differences and consequences Joakim Nyberg, Sebastian Uecker - PowerPoint PPT Presentation

Approximations of the population Fisher information matrix- differences and consequencesJoakim Nyberg, Sebastian Ueckert, Andrew C. Hooker

• At PODE 2009 all Population Optimal Design (OD)
• Software should evaluate the same simple Warfarin problem…
• 1-compartment model, 1st order absorption, oral dose 70 mg
• Proportional error model (σ2=0.01)
• 32 subjects with 8 measurements at

0.5, 1, 2, 6 ,24, 36, 72,120 hours (evaluation)

FIM can be calculated in different ways:

Assuming var(y) w.r.t. the fixed effects=0

Assuming var(y) w.r.t. the fixed effects≠0

A* is somewhat modified/updated if full is used, i.e.

Different between

full and reduced

The FIM

• If we have correlation between fixed effects
• and random effects in like the FULL is the “theoretically
• correct” method.
• If not, the Reduced is “correct theoretically”

but this is seldom the case in Pharmacometrics

The “truth”

*

*

* Retout, Mentré – Further developments of FIM in NLME-models…. J. BioPharm. Stat 2003

The “truth”

Possibly issues with the

Cramér-Rao inequality

• Software gave similar results with similar approximations
• Reduced superior to Full in terms of predicting the “truth”
• Even less predictive performance with higher order

FOCE-based FIM.

• The derivation of Full or Red is wrong
• Derive FIM with simulations, i.e. integrate over observed FIM
• FO-approximation too poor

- FOCE is obviously not enough, try high order approximations

• Asymptotic behavior (FIM-1≤COV)

- Increase data set x 2 => SE should decrease by 2(1/2)

• Numerical instability in Full but not in Red FIM

- Using automatic differentiation (AD) to avoid step length issues

• Estimation software is not true ML-estimator,
• i.e. efficiency of estimator not accurate
• - NM hard to know how the parameter search is performed
• but Monolix well documented

• ln-transform model to have additive res-error

(avoiding interaction terms)

• Check that the problem holds for prop IIV structure

(FO approximation => proportional IIV = exp IIV)

• Fix all parameters except fixed effect Ka

ln model, add error, exp IIV = prop IIV

• Issues still remaining => work with simplified model

* 100 000 bootstrap samples

** Retout, Mentré – Further developments of FIM in NLME-models…. J. BioPharm. Stat 2003

• Asymptotic behavior (FIM-1≤COV)
• Increase data set x 2 => SE should decrease by 2(1/2)
• Numerical instability in Full but not in Red FIM

- Using automatic differentiation (AD) to avoid step length issues

None of this affected the results (2 down 3 to go)

• The derivation of Full or Red is wrong
• Derive FIM with simulations, i.e. integrate over observed FIM
• FO-approximation too poor

- FOCE is obviously not enough, try high order approximations

SO

Similar

SO – Closer to truth

Simulation based FIM FOCE from NONMEM solve the problem!

* 100 000 bootstrap samples

• The derivation of Full or Red is wrong
• Integration FIM ≈ Analytic FIM => Not the answer
• FO-approximation too poor
• SO shrinks the differences but still to poor of an approx.
• FOCE is worse but NONMEM integrated FOCE FIM is good?!
• Possibly issues with the FOCE method?

• NONMEM FOCE assumes linearization around the mode
• of the distribution => correlation between the individual
• parameters and the population parameters.
• Analytic FIMFOCE * does not assume this
• To calculate individual mode data is needed

• Update Analytic FIMFOCE to include the correlation:
• Calculate Expected Empirical Bayes Estimates (EEBE)

EEBE are not data dependent

• Whenever PopED differentiates pop parameters;

differentiate EEBE as well

* Retout, Mentré – Further developments of FIM in NLME-models…. J. BioPharm. Stat 2003

• The new FOCE method solves the problem!

• The Full FIM does not always work

with the FO-approximation

Full

Reduced

6

2

2

2

1

1

1

1

3 support points

5 support points

Surface of |FIM| for SOCE-MC

SOCE>MC

SOCE=MC

SOCE<MC

Optimal MC

Optimal SOCE

Example from PAGE 2009 Nyberg et al

Full |FIM|

Reduced |FIM|

Similar results with the transit compartment model

Results from Nyberg et al PODE 2008

Full Ka RSE(%)

Reduced Ka RSE(%)

S3-S7: (7.79h)

S3-S7: (7.79h)

S8: (120h)

S8: (120h)

Full ≈ Red

• The FO-approximation is not always enough for Full FIM

Possibly also too poor approx. for Reduced

• Reduced FIM collapses occasionally
• High order approximations stabilize differences
• Different approximations give different optimal designs,

e.g. different sampling times and different number of

support points

• If runtime allows – Use high order approximations

FOCE, SO, SOCE, MC etc.

• If Red is stable – Use reduced to optimize but evaluate with both
• If Red is unstable – Optimize with Full but evaluate with Red

Beware: - No “golden” solution is presented

- The Cramer-Rao inequality does not hold comparing

different methods when optimizing / estimating

- To get “correct” SE from the FIM either sim/est needs

to be performed or high order FIMs need to be evaluated

I would like to acknowledge Sergei Leonov for our interesting emails

discussing these issues.